BackgroundPart I: Martingale Problem Under Sublinear ExpectationPart II: Ito’s calculas in the sublinear expectation space
Summary
Ito’s Calculas under Sublinear Expectations viaRegularity of PDEs and Rough Path
Xin Guo
UC Berkeley
METE Conference, June 8, 2018
Based on the joint work with Chen Pan (NUS)
Guo & Pan Ito’s Calculas under Sublinear Expectations via Regularity of PDEs and Rough Path
BackgroundPart I: Martingale Problem Under Sublinear ExpectationPart II: Ito’s calculas in the sublinear expectation space
Summary
Work related to Mete
1. W. H. Fleming and H. M. Soner (1992), Controlled Markov Processesand Viscosity Solutions, Springer-Verlag, New York.2. Cheridito, H. M. Soner, N. Touzi, and N. Victoir (2007), Second-orderbackward stochastic differential equations and fully nonlinear parabolicPDEs, Communications on Pure and Applied Mathematics, LX,1081-1110.3. H. M. Soner, N. Touzi, and J. F. Zhang (2011), Martingalerepresentation theorem for the G -expectation, Stochastic Processes andtheir Applications, 121(2), 265-287.4. H. M. Soner, N. Touzi, and J. F. Zhang (2011), Quasi-sure stochasticanalysis through aggregation, Electronic Journal of Probability, 16(67),1844-1879.5. H. M. Soner, N. Touzi, and J. F. Zhang (2012), Well-posedness ofsecond order backward SDEs, Probability Theory and RelatedFields,153(1-2), 149-190.
Guo & Pan Ito’s Calculas under Sublinear Expectations via Regularity of PDEs and Rough Path
BackgroundPart I: Martingale Problem Under Sublinear ExpectationPart II: Ito’s calculas in the sublinear expectation space
Summary
Outline
1 Background
2 Part I: Martingale Problem Under Sublinear Expectation
3 Part II: Ito’s calculas in the sublinear expectation space
4 Summary
Guo & Pan Ito’s Calculas under Sublinear Expectations via Regularity of PDEs and Rough Path
BackgroundPart I: Martingale Problem Under Sublinear ExpectationPart II: Ito’s calculas in the sublinear expectation space
Summary
Linearity in probability theory
1-1 correspondence between linear expectation and additiveprobability measure,
P(X ∈ A) = E [1A].
Guo & Pan Ito’s Calculas under Sublinear Expectations via Regularity of PDEs and Rough Path
BackgroundPart I: Martingale Problem Under Sublinear ExpectationPart II: Ito’s calculas in the sublinear expectation space
Summary
Sublinear Expectation E
Given χ (e.g. all bounded measurable random variables), E : χ→ R issublinear iff
(a) Monotonicity: If X ≤ Y , then E [X ] ≤ E [Y ]
(b) Constant preserving: E [X + c] = E [X ] + c
(c) Sublinearity: E [X + Y ] ≤ E [X ] + E [Y ].
(d) Positive homogeneity: E [λX ] = λE [X ] for all λ > 0
Guo & Pan Ito’s Calculas under Sublinear Expectations via Regularity of PDEs and Rough Path
BackgroundPart I: Martingale Problem Under Sublinear ExpectationPart II: Ito’s calculas in the sublinear expectation space
Summary
No more 1-1 correspondence between E and P
ClearlyP(A) = E [1A] = Ef [1A]
for all f continuous and strictly increasing, f (x) = x for x ∈ [0, 1], whereEf [X ] = f −1(E [f (X )]).
Guo & Pan Ito’s Calculas under Sublinear Expectations via Regularity of PDEs and Rough Path
BackgroundPart I: Martingale Problem Under Sublinear ExpectationPart II: Ito’s calculas in the sublinear expectation space
Summary
Linear and sublinear expectations (Denis, Hu and Peng (2011))
There exists a weakly compact family of probability measures P such that
E [X ] = maxP∈P
EP [X ],
where EP is the linear expectation with respect to P, for a proper classof random process X .
Sublinear expectation and model uncertainty
Sublinear expectation “measures” the model uncertainty, the bigger theexpectation E , the more the uncertainty.
E1[X ] ≤ E2[X ] iff P1 ⊂ P2
Guo & Pan Ito’s Calculas under Sublinear Expectations via Regularity of PDEs and Rough Path
BackgroundPart I: Martingale Problem Under Sublinear ExpectationPart II: Ito’s calculas in the sublinear expectation space
Summary
Sublinear expectation and risk measure
Letρ(X ) = E [−X ]
Then we get a coherent risk measure ρ : χ→ R
(a) Monotonicity: If X ≥ Y , then ρ(X ) ≤ ρ(Y ).
(b) Constant translatability: ρ(X + c) = ρ(X )− c
(c) Convexity: ρ(αX + (1− α)Y ) ≤ αρ(X ) + (1− α)ρ(Y ),α ∈ [0, 1].
(d) Positive homogeneity: ρ(λX ) = λρ(X ) for all λ > 0.
Guo & Pan Ito’s Calculas under Sublinear Expectations via Regularity of PDEs and Rough Path
BackgroundPart I: Martingale Problem Under Sublinear ExpectationPart II: Ito’s calculas in the sublinear expectation space
Summary
G - expectation theory (Peng (2005))
G -normal distribution N(0× [σ2, σ2]), characterized by the G -heatequation
∂tu − G (D2u) = 0, u|t=0 = φ.
Here G (·) : R→ R is a monotonic, sublinear function, with
G (γ) =1
2sup
α∈[σ2,σ2]
γα,
G− Brownian Motion, via the notion of G -independence
Ito’s calculus developed under G -Ito’s isometry
Guo & Pan Ito’s Calculas under Sublinear Expectations via Regularity of PDEs and Rough Path
BackgroundPart I: Martingale Problem Under Sublinear ExpectationPart II: Ito’s calculas in the sublinear expectation space
Summary
Link between PDEs and probability
Independence: probability= measure theory + notion ofindependence
Regularity: the subject of PDEs/Control
Stroock and Varadhan (1979) explored the regularity of the solutionto a linear PDE for the uniqueness of the solution to a martingaleproblem
Guo & Pan Ito’s Calculas under Sublinear Expectations via Regularity of PDEs and Rough Path
BackgroundPart I: Martingale Problem Under Sublinear ExpectationPart II: Ito’s calculas in the sublinear expectation space
Summary
Our work
Propose and study the martingale problem in a sublinear expectationspace in the spirit of Stroock and Varadhan (1979)
Build Ito’s calculus for the canonical process in this sublinearexpectation space
Key ideas: Regularity of fully nonlinear PDEs and rough path theory
Guo & Pan Ito’s Calculas under Sublinear Expectations via Regularity of PDEs and Rough Path
BackgroundPart I: Martingale Problem Under Sublinear ExpectationPart II: Ito’s calculas in the sublinear expectation space
Summary
Martingale problem
Find a family of operators Ett≥0 on a sublinear expectation space(Ω,H) such that
ϕ(Xt)−∫ t
0
G (Xθ, ϕx(Xθ), ϕxx(Xθ)) dθ, t ≥ 0
is an Et-martingale for all ϕ ∈ C∞0 (Rd).
G : Rd × Rd × Sd → R continuous with desirable properties
Ω = Cx0 ([0,∞);Rd) and Xt(ω) = ω(t), ω ∈ Ω.
Guo & Pan Ito’s Calculas under Sublinear Expectations via Regularity of PDEs and Rough Path
BackgroundPart I: Martingale Problem Under Sublinear ExpectationPart II: Ito’s calculas in the sublinear expectation space
Summary
Our martingale problems vs classical martingale problems
Classical M.P.’s Our M.P.’s
to find a probability measure Pon (Ω,F)
to find a sublinear expectation E on(Ω,H)
X0 = x P-a.s. X0 = x in L1E or L2
Eϕ(Xt) −
∫ t
0Lθϕ(Xθ) dθ is a P-
martingale for ∀ϕ ∈ C∞0
ϕ(Xt) −∫ t
0G (Xθ, ϕx(Xθ), ϕxx(Xθ))dθ is
an E-martingale for ∀ϕ ∈ C∞0Lθ = 1
2
∑aij(θ, ·) ∂2
∂xi∂xj+∑
bi (θ, ·) ∂∂xi
is a linear differen-tial operator
Nonlinear PDE associated with G : Rd ×Rd × Sd → R is a continuous functionwith some properties
Guo & Pan Ito’s Calculas under Sublinear Expectations via Regularity of PDEs and Rough Path
BackgroundPart I: Martingale Problem Under Sublinear ExpectationPart II: Ito’s calculas in the sublinear expectation space
Summary
Choice of G
A function G : Rd × Rd × Sd → R
G = G (x , p,A) = supγ∈Γ
1
2tr[a(x , γ)A] + (b(x , γ), p)
,
Γ given compact metric space
a(x , γ) = σ(x , γ)σ′(x , γ) positive semidefinite
σ, b ∈ C (Rd × Γ), and σ and b uniformly bounded and uniformlyLipschitz continuous with respect to x
Guo & Pan Ito’s Calculas under Sublinear Expectations via Regularity of PDEs and Rough Path
BackgroundPart I: Martingale Problem Under Sublinear ExpectationPart II: Ito’s calculas in the sublinear expectation space
Summary
This function G : Rd × Rd × Sd → R satisfies
A. Subadditivity G (x , p + p,A + A) ≤ G (x , p,A) + G (x , p, A);
B. Positive Homogeneity G (x , λp, λA) = λG (x , p,A);
C. Monotonicity G (x , p,A) ≤ G (x , p,A + A);
D. G is uniformly Liptschitz continuous with respect to x .
Guo & Pan Ito’s Calculas under Sublinear Expectations via Regularity of PDEs and Rough Path
BackgroundPart I: Martingale Problem Under Sublinear ExpectationPart II: Ito’s calculas in the sublinear expectation space
Summary
PDEs associated with G
For a given G , the associated state-dependent parabolic PDE∂tu(t, x)− G (x ,Du(t, x),D2u(t, x)) = 0, (t, x) ∈ (0,∞]× Rd ,
u(0, x) = ϕ(x), x ∈ Rd .
a(·, ·) ≡ I and b(·, ·) ≡ 0, this PDE (P) is the heat equation
In general, this type of PDE (P) corresponds to the HJB equationfrom stochastic control
Guo & Pan Ito’s Calculas under Sublinear Expectations via Regularity of PDEs and Rough Path
BackgroundPart I: Martingale Problem Under Sublinear ExpectationPart II: Ito’s calculas in the sublinear expectation space
Summary
Known results about related PDEs
Fleming and Soner (1992)
There exists a unique viscosity solution for this PDE (P) with polynomialgrowth, assuming ϕ(x) ∈ C (Rd) with a polynomial growth.
Evans-Krylov (1982)
For a class of convex, fully nonlinear PDEs, if ϕ(x) ∈ Cb([0,T ]× Rd),then the solution u possess the following properties:
I) u ∈ Cb([0,T ]× Rd);
II) there exists a constant α0 ∈ (0, 1) only depending on d ,K , ε suchthat for each κ > 0, ‖u‖C 2+α0 ([κ,T ]×Rd ) <∞.
Furthermore, if ϕ ∈ C 2+α1 (Rd) and bounded, thenu ∈ C 2+α([0,T ]× Rd) for α = α0 ∧ α1 ∈ (0, 1).
Guo & Pan Ito’s Calculas under Sublinear Expectations via Regularity of PDEs and Rough Path
BackgroundPart I: Martingale Problem Under Sublinear ExpectationPart II: Ito’s calculas in the sublinear expectation space
Summary
Properties of such PDEs
By the property of G , the stability of the viscosity solution andEvans-Krylov (1982),
Smooth approximation of viscosity solutions
If ϕ(x) ∈ C∞0 (Rd) The unique solution of the PDE can be approximatedby C 2+α solutions of the same type of PDEs on compact subsets.
Properties of solutions to the PDEs
Let uϕ ∈ C ([0,T ]× Rd) denote the unique solutions of (P) withpolynomial growth, respectively. Then
uϕ+c = uϕ + c ,
uϕ − uφ ≤ uϕ−φ,
uλϕ = λuϕ, λ ≥ 0.
with c ∈ R a constant, and ϕ, φ continuous with polynomial growth.
Guo & Pan Ito’s Calculas under Sublinear Expectations via Regularity of PDEs and Rough Path
BackgroundPart I: Martingale Problem Under Sublinear ExpectationPart II: Ito’s calculas in the sublinear expectation space
Summary
Constructing conditional expectations
Finite dimensional construction via backward induction, in the spiritof BSDE by using the unqiue solution of the PDEs
Extend to a Banach space under the norm ‖ · ‖1 = E [| · |] or‖ · ‖2 =
√E [| · |2].
Guo & Pan Ito’s Calculas under Sublinear Expectations via Regularity of PDEs and Rough Path
BackgroundPart I: Martingale Problem Under Sublinear ExpectationPart II: Ito’s calculas in the sublinear expectation space
Summary
The finite dimensional construction
Ω = Cx0 ([0,∞);Rd), Xt(ω) = ωt ∈ Ω is canonical
Define an operator Tt [ϕ(·)](x) = u(t, x), where u is the uniqueviscosity solution of the PDE
Take ϕ in a proper function space, say Cl,Lip((Rd)N) whereCl.Lip(Rn) is the space of real-valued continuous functions defined onRn such that for all ϕ ∈ Cl.Lip(Rn),
|ϕ(x)− ϕ(y)| ≤ C (1 + |x |m + |y |m)|x − y |, ∀ x , y ∈ Rn
for some C > 0,m ∈ N depending on ϕ.
Guo & Pan Ito’s Calculas under Sublinear Expectations via Regularity of PDEs and Rough Path
BackgroundPart I: Martingale Problem Under Sublinear ExpectationPart II: Ito’s calculas in the sublinear expectation space
Summary
Set ξ(ω) = ϕ(Xt1 , · · · ,XtN ), 0 = t0 ≤ t1 ≤ · · · ≤ tN ≤ T ,
Define Et by
Et [ξ] = ϕN−j(ωt1 , · · · , ωtj ), if t = tj , 0 ≤ j ≤ N,
Here
ϕ1(x1, · · · , xN−1) = TtN−tN−1[ϕ(x1, · · · , xN−1, ·)](xN−1),
. . .
ϕN−j(x1, · · · , xj) = Ttj+1−tj [ϕN−j−1(x1, · · · , xj , ·)](xj),
. . .
ϕN−1(x1) = Tt2−t1 [ϕN−2(x1, ·)](x1),
ϕN = Tt1 [ϕN−1(·)](x0),
Guo & Pan Ito’s Calculas under Sublinear Expectations via Regularity of PDEs and Rough Path
BackgroundPart I: Martingale Problem Under Sublinear ExpectationPart II: Ito’s calculas in the sublinear expectation space
Summary
Sublinear expectation space (Ω,H, E)
(ΩT ,HT , E) is a sublinear expectation space, withΩT := ω·∧T ;ω ∈ Ω andHT := Lip(ΩT ) = ϕ(Xt1 , · · · ,XtN );ϕ ∈ Cl.Lip((Rd)N) for someN ∈ N and 0 ≤ t1 ≤ · · · ≤ tN ≤ TFor each t ∈ [0,T ], one can extend the spaceLip(Ωt) ⊂ Lip(ΩT ), t ≤ T to a Banach space LiE(Ωt) under the norm
‖ · ‖1 = E [| · |] or ‖ · ‖2 =√E [| · |2]. Set H = LiE := ∪T≥0L
iE(ΩT )
Guo & Pan Ito’s Calculas under Sublinear Expectations via Regularity of PDEs and Rough Path
BackgroundPart I: Martingale Problem Under Sublinear ExpectationPart II: Ito’s calculas in the sublinear expectation space
Summary
Properties of EGiven such a sublinear expectation space (Ω,H, E). For any ξ, η ∈ H
(Monotonicity)Et [ξ] ≤ Et [η] if ξ ≤ η
(Constant preserving) For c ∈ R constant,
E [ξ + c] = E [ξ] + c
(Tower property)
Es Es+h = Es , h > 0.
(Subadditivity) Et [ξ + η] ≤ Et [ξ] + Et [η].(Positive homogeneity)
Es [ξη] = ξ+Es [η] + ξ−Es [−η].
In particular,Et [λξ] = λEt [ξ]
for any constant λ ≥ 0.For the time dependent G , the tower property becomes the semi-groupproperty.
Guo & Pan Ito’s Calculas under Sublinear Expectations via Regularity of PDEs and Rough Path
BackgroundPart I: Martingale Problem Under Sublinear ExpectationPart II: Ito’s calculas in the sublinear expectation space
Summary
Martingale Problem
Let Ω = Cx([0,∞);Rd) with ω0 = x ∈ Rd , set Xt(ω) := ωt . Given thePDE (P) with ϕ ∈ C∞0 (R).
ϕ(Xt)−∫ t
0
G (Xθ, ϕx(Xθ), ϕxx(Xθ)) dθ] = 0, 0 ≤ s ≤ t <∞.
is an E− martingale. Here E = Et with Et the family of conditionalexpectations on the sublinear expectation space generated from the PDE(P).
Guo & Pan Ito’s Calculas under Sublinear Expectations via Regularity of PDEs and Rough Path
BackgroundPart I: Martingale Problem Under Sublinear ExpectationPart II: Ito’s calculas in the sublinear expectation space
Summary
E-Martingale
Given a sublinear expectation space (Ω,H, E), a stochastic process(ξt)t≥0 is a collection of random variables on (Ω,H), i.e., for each t ≥ 0,ξt ∈ LiE(Ωt), i = 1, 2.A stochastic process (Mt)t≥0 is called an E-martingale if for eacht ∈ [0,∞),Mt ∈ LiE(Ωt), and for each s ∈ [0, t],
Es [Mt ] = Ms .
Guo & Pan Ito’s Calculas under Sublinear Expectations via Regularity of PDEs and Rough Path
BackgroundPart I: Martingale Problem Under Sublinear ExpectationPart II: Ito’s calculas in the sublinear expectation space
Summary
Key idea to solving the martingale problem
The positive homogeneity, monotonicity, and the constant preservingof E leads to
Given a sublinear expectation space (Ω,H, E). Let X ∈ H be given.Then for each sequence ϕn∞n=1 ⊂ C(Rd) satisfying ϕn ↓ 0, we haveE [ϕn(X )] ↓ 0.
Focus on the simple process and smooth solutions of PDE.Take π = θi ; θ0 = s < θ1 < · · · < θK = t,K ∈ N, ‖π‖ =max1≤k≤K |∆θk |,∆θk = θk − θk−1
g(Xθ) := G (Xθ, ϕx(Xθ), ϕxx(Xθ)).Given a smooth PDE solution in the sense of Evans-Krylov,
Es [K∑
k=1
ϕ(Xθk )− ϕ(Xθk−1)− g(Xθk−1
)∆θk] = o(1), as ‖π‖ → 0.
Guo & Pan Ito’s Calculas under Sublinear Expectations via Regularity of PDEs and Rough Path
BackgroundPart I: Martingale Problem Under Sublinear ExpectationPart II: Ito’s calculas in the sublinear expectation space
Summary
The proof needs ∂tu to be uniformly α2 -Holder continuous in t, because
Eθk−1[ϕ(Xθk )− ϕ(Xθk−1
)− g(Xθk−1)∆θk ]
=Eθk−1[ϕ(Xθk )]− ϕ(Xθk−1
)− g(Xθk−1)∆θk
= u(∆θk ,Xθk−1)− u(0,Xθk−1
)− g(Xθk−1)∆θk
=
[∂u
∂t(0,Xθk−1
)− g(Xθk−1)
]∆θk + O((∆θk)1+ α
2 )
= O((∆θk)1+ α2 )
Guo & Pan Ito’s Calculas under Sublinear Expectations via Regularity of PDEs and Rough Path
BackgroundPart I: Martingale Problem Under Sublinear ExpectationPart II: Ito’s calculas in the sublinear expectation space
Summary
Solution of martingale problem leads to
Key moment estimates
Given T > 0. Let t ∈ [0,T ] and h > 0 such thatti = t + ih ∈ [0,T ], i = 0, 1, 2, 3. Then we have the following estimates
|Et [±(Xt − Xs)]| ≤ L|t − s|,Et [(Xt+h − Xt)
2n] ≤ C (n, L,T )hn,
Et [±(Xt+2h − Xt+h)(Xt+h − Xt)]| ≤ L√
C (1, L,T )h32 ,
Et [|(Xt1 − Xt0 )(Xt2 − Xt1 )(Xt3 − Xt2 )|] ≤ C (1, L,T )32 h
32 ,
where L is the Lipschitz constant of G , C (n, L,T ) is a constantdepending only on n, L and T .
Guo & Pan Ito’s Calculas under Sublinear Expectations via Regularity of PDEs and Rough Path
BackgroundPart I: Martingale Problem Under Sublinear ExpectationPart II: Ito’s calculas in the sublinear expectation space
Summary
Remark
Our estimate is consistent with the G -framework where independentincrements assumption is needed.Take Xt = Bt + 〈B〉t , then under the G -expectation E,
E[(X2h − Xh)Xh] = E[(B2h + 〈B〉2h − Bh − 〈B〉h)(Bh + 〈B〉h)]
∼O(h32 ).
The moment estimates are crucial for establishing Ito’s calculus asthe random process in the sublinear expectation space constructedfrom the PDE in general has no independent increment
Guo & Pan Ito’s Calculas under Sublinear Expectations via Regularity of PDEs and Rough Path
BackgroundPart I: Martingale Problem Under Sublinear ExpectationPart II: Ito’s calculas in the sublinear expectation space
Summary
Moments estimates leads to
〈X 〉tL2E= lim
n→∞
∑ΠN
|Xtk+1− Xtk |2,
∫ T
0
φ(Xt) dXtL2E= lim‖ΠN‖→0
∑tk∈ΠN
φ(Xtk )(Xtk+1− Xtk ),
with ΠN = tk ; tk = kt/N, k = 0, 1, . . . ,N
Guo & Pan Ito’s Calculas under Sublinear Expectations via Regularity of PDEs and Rough Path
BackgroundPart I: Martingale Problem Under Sublinear ExpectationPart II: Ito’s calculas in the sublinear expectation space
Summary
For general φ(X ) and general partition
To show the limit is independent of the partition, need “sewing lemma”for estimating error term.To see this, assume φ ∈ C 1(R) for each u ∈ [s, t], and approximateφ(Xu) by
φ(Xu) ≈ φ(Xs) + φ′(Xs)Xs,u.
Then formally the integral∫ T
0
φ(Xθ) dXθ ≈∑ΠN
[φ(Xtk )Xtk ,tk+1
+ φ′(Xtk )
∫ tk+1
tk
Xtk ,θ dXθ
],
with the error term
R(s, t) :=
∫ t
s
φ(Xθ) dXθ −[φ(Xs)Xs,t + φ′(Xs)
∫ t
s
Xs,θ dXθ
]Establishing “Sewing Lemma” to estimate
‖R(s, t)‖2 :=√E [|R(s, t)|2] ≤ C |t − s|β , for some β > 1
Guo & Pan Ito’s Calculas under Sublinear Expectations via Regularity of PDEs and Rough Path
BackgroundPart I: Martingale Problem Under Sublinear ExpectationPart II: Ito’s calculas in the sublinear expectation space
Summary
Generalized Ito’s isometry
Suppose ϕ ∈ C 1(R) and ϕ′ ∈ P. Then
Es
(∫ s+h
s
ϕ(Xθ) dXθ
)2 ≤ C1h, (1)
Es
(∫ s+h
s
ϕ(Xθ) dXθ
)4 ≤ C2h
2, (2)
Es
[±
(∫ s+h
s
ϕ(Xθ) dXθ
)]≤ C3h, (3)
with h > 0, s, s + h ∈ [0,T ], and C1,C2,C3 depending on ϕ, L, and T ,
P = ϕ ∈ C (R) : there exist constants C > 0, p ∈ N, γ ∈ (0, 1] s.t.
|ϕ(x)− ϕ(y)| ≤ C (1 + |x |p + |y |p)|x − y |γ for any x , y ∈ R.
Guo & Pan Ito’s Calculas under Sublinear Expectations via Regularity of PDEs and Rough Path
BackgroundPart I: Martingale Problem Under Sublinear ExpectationPart II: Ito’s calculas in the sublinear expectation space
Summary
Ito’s formula
Let
ηt = η0 +
∫ t
0
µ(Xθ) dθ +
∫ t
0
ρ(Xθ) d〈X 〉θ +
∫ t
0
ς(Xθ) dXθ, 0 ≤ t ≤ T ,
where η0 ∈ R is a constant. Suppose ς ∈ C 1(R) and ς ′ ∈ P. Then, forany f ∈ C 2(R) satisfying f ′′ ∈ P,
f (ηt) = f (η0) +
∫ t
0
f ′(ηθ) dηθ +1
2
∫ t
0
f ′′(ηθ)ς2(Xθ) d〈X 〉θ.
Guo & Pan Ito’s Calculas under Sublinear Expectations via Regularity of PDEs and Rough Path
BackgroundPart I: Martingale Problem Under Sublinear ExpectationPart II: Ito’s calculas in the sublinear expectation space
Summary
Integration by parts
Let φ, ψ ∈ C 1(R), and φ′, ψ′ ∈ P, then under the L2E norm
φ(Xt)ψ(Xt)− φ(Xs)ψ(Xs) =
∫ t
s
φ(Xθ) dψ(Xθ)
+
∫ t
s
ψ(Xθ) dφ(Xθ) + 〈φ(X ), ψ(X )〉s,t
Here the cross variation process 〈φ(X ), ψ(X )〉 is defined by
〈φ(X ), ψ(X )〉 =1
4[〈φ(X ) + ψ(X )〉 − 〈φ(X )− ψ(X )〉]
and the quadratic variation 〈φ(X )〉t defined as
〈φ(X )〉t = φ(Xt)2 − φ(X0)2 − 2
∫ t
0
φ(Xθ) dφ(Xθ) in L2E .
Guo & Pan Ito’s Calculas under Sublinear Expectations via Regularity of PDEs and Rough Path
BackgroundPart I: Martingale Problem Under Sublinear ExpectationPart II: Ito’s calculas in the sublinear expectation space
Summary
Summary
A martingale problem under sub-linear expectation is proposed andstudied, and associated Ito’s calculus is developed
The analytical properties of PDE, especially the semi-group propertyand the regularity of the solution, replace the dependence structureof the random process X , and leads to generalized Ito’s isometry.
Extension to path dependent PDEs or jump processes?
Guo & Pan Ito’s Calculas under Sublinear Expectations via Regularity of PDEs and Rough Path
BackgroundPart I: Martingale Problem Under Sublinear ExpectationPart II: Ito’s calculas in the sublinear expectation space
Summary
More related works
Construction of stochastic integrals with respect to semi-martingale:Bichleter (1981), Nutz (2012),
G-SDEs with rough paths: Geng, Qian, and Yang (2014), Peng andZhang (2015)
Rough path: Lyons (1998), Friz and Hairer (2014)
Guo & Pan Ito’s Calculas under Sublinear Expectations via Regularity of PDEs and Rough Path
BackgroundPart I: Martingale Problem Under Sublinear ExpectationPart II: Ito’s calculas in the sublinear expectation space
Summary
Reference
Cheridito, H. M. Soner, N. Touzi, and N. Victoir (2007), Second-order backward stochastic differential equations and fullynonlinear parabolic PDEs, Communications on Pure and Applied Mathematics,LX, 1081-1110.
L. C. Evans (1982), Classical solutions of fully nonlinear, convex, second-order elliptic equations, Communications in Pure andApplied Mathematics, 35(3), 333-363.
D. Feyel and A. de La Pradelle (2006), Curvelinear integrals along enriched paths, Electronic Journal of Probability 16(34),860-892.
P. Friz and M. Hairer (2014), A Course on Rough Paths – with an Introduction to Regularity Structures, Springer, Berlin.
W. H. Fleming and H. M. Soner (1992), Controlled Markov Processes and Viscosity Solutions, Springer-Verlag, New York.
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Guo & Pan Ito’s Calculas under Sublinear Expectations via Regularity of PDEs and Rough Path
BackgroundPart I: Martingale Problem Under Sublinear ExpectationPart II: Ito’s calculas in the sublinear expectation space
Summary
u ∈ Cα(Q), α ∈ (0, 1], if
‖u‖Cα(Q) := supx 6=y ,x,y∈Rd
s 6=t,s,t∈[0,T ]
|u(t, x)− u(s, y)|(|t − s| 12 + |x − y |)α
<∞.
u ∈ C 2+α(Q), α ∈ (0, 1], if
‖u‖C(Q)+‖∂tu‖C(Q)+‖Du‖C(Q)+‖D2u‖C(Q)+‖∂tu‖Cα(Q)+‖D2u‖Cα(Q) < ∞.
C∞0 (Rd) is the space of C∞ functions having compact support onRd .
Guo & Pan Ito’s Calculas under Sublinear Expectations via Regularity of PDEs and Rough Path
BackgroundPart I: Martingale Problem Under Sublinear ExpectationPart II: Ito’s calculas in the sublinear expectation space
Summary
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Guo & Pan Ito’s Calculas under Sublinear Expectations via Regularity of PDEs and Rough Path